pythagorian fuzzy matrices

Proceedings of the KMA National Conference on Recent Trends in Discrete and Fuzzy Mathematics, Bharata Mata College, Thrikkakara, Kochi-21, Kerala; Nov. 10-12, 2005, Pages 61-63.

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PYTHAGOREAN FUZZY MATRICES A. NAGOOR GANI* and G. KALYANI†

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ABSTRACT. In this paper, a non-negative quantity m-norm is introduced on each and every

fuzzy matrix in Fn where Fn is the set of all n×n fuzzy matrices. Based on this m-norm Pythagorean triplets are formed on Fuzzy matrices. It is also shown that the set Pij of all

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Pythagorean triplets form a commutative monoid. In the same way, Pythagorean Quadruples

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in Fuzzy matrices and progressional fuzzy matrices are also defined.

1. INTRODUCTION

We consider Fn, the set of all (n×n) matrices over F=[0,1]. We define in F the following operation.

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(a1 + a2 + … + an )k = a1k + a2k + … + ank

∀k≥1

We define ‘+’ and scalar multiplication in Fn as A+B = [aij + bij] where A = [aij] and B = [bij] and cA = [caij] where c is in [0, 1].

In this paper a subset of Fn is taken and based on the above operations and the introduction of the

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following m-norm [3], Pythagorean fuzzy matrices are defined and some of its properties are seen. 1.1 Definition: For every A in Fn the m-norm of A is defined as ||A||m = max [aij], where A = [aij]. Consider a subset of Fn, which is such that all the elements in that subset have the same maximum

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entry position. That is for all matrices in the subset the m-norm positions are the same. Thus Fij is a subset of Fn such that for each matrix in Fij the (i, j) th position is the maximum entry position. Thus there are n2 collections, which are denoted by F11, F12, . . ., Fnn, where Fi j= {P = [pij] ∈ Fn : ||P||m = pij}, for i, j = 1 to n.

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1.2 Definition: Consider the collection Fij. If three matrices in Fij are such that their m-norms satisfy the Pythagorean law, then the three matrices are known as Pythagorean triplet in Fn.

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Note: There are so many Pythagorean triplets (A, B, C) in Fij satisfying the same law c2= a2+ b2. 1.3 Example: If A =

⎡ 0.3 0 ⎤ ⎡ 0.4 0.2 ⎤ ⎡ 0.5 0.1⎤ 2 2 2 ⎢⎣ 0.1 0.2 ⎥⎦ B = ⎢⎣ 0.1 0.3 ⎥⎦ & C = ⎢⎣ 0 0.4 ⎥⎦ , then c = a + b since c =

0.5, a = 0.3, b = 0.4.

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P.G. & Research Department of Mathematics, Jamal Mohamed College, Tiruchirappalli, Tamil Nadu.

†

Department of Mathematics, Seethalakshmi Ramaswami College (Autonomous), Tiruchirappalli, Tamil Nadu.

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PYTHAGOREAN FUZZY MATRICES

If A =

⎡ 0.3 0.2 ⎤ ⎢⎣ 0.1 0.1⎦⎥

B=

⎡ 0.4 0 ⎤ ⎡ 0.5 0.4 ⎤ ⎢⎣ 0.1 0.2 ⎥⎦ & C = ⎢⎣ 0.3 0.2 ⎥⎦ , then this (A, B, C) also satisfies the

law c2 = a2 + b2.

⎡ 0.3 0 ⎤ ⎢⎣ 0.2 0.1⎥⎦

B=

⎡ 0.4 0.1⎤ ⎡ 0.5 0.3⎤ ⎢⎣ 0.2 0.2 ⎥⎦ & C = ⎢⎣ 0.3 0.1⎥⎦ and many cases like this.

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Similarly A =

Clearly, these matrices belong to the collection F11.

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1.4 Example: Let a = 0.05, b = 0.12 and c = 0.13. Clearly c2 = a2+ b2. Let Pij = {(A, B,C) | c2 = a2 + b2}, the collection of all Pythagorean triplets. Clearly, Pij ⊆ Fij × Fij × Fij ⊆ Fn × Fn × Fn. We can show that Pij forms a commutative monoid for i = 1 to n, j = 1 to n with respect to the following addition:

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(A1, B1, C1) + (A2, B2, C2) = (A1+A2, B1+B2, C1+C2)

Clearly, ||C1 + C2 ||m2 = (c1 + c2)2 = c12 + c22 = a12 + b12 + a22 + b22 = a12 + a22 + b12+ b22 Pij, then (A1+A2, B1+B2, C1+C2) also belongs to Pij.

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= (a1 + a2)2 + (b1+ b2)2 = ||A1 + A2 ||m2 + ||B1 + B2 ||m2. Thus, whenever (A1, B1, C1) & (A2, B2, C2) are in 1.5 Theorem: Pij is a commutative monoid with respect to the above addition. Proof : By the above, Pij is closed with respect to the addition defined. Commutativity of matrix

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addition implies (A1, B1, C1) + (A2, B2, C2) = (A2, B2, C2) + (A1, B1, C1); for, (A1+A2, B1+B2, C1+C2) = (A2 + A1, B2 + B1, C2 + C1).

Also, (A1, B1, C1) + {(A2, B2, C2) + (A3, B3, C3)} = {(A1, B1, C1) + (A2, B2, C2)} + (A3, B3, C3) = (A1+A2+A3, B1+B2+B3, C1+C2+C3), since (c1 + c2 + c3)2 = c12 + c22+ c32 = a12 + b12 + a22 + b22 + a32 +

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b32 = a12 + a22 + a32 + b12+ b22+ b32 = (a1 + a2 + a3)2 + (b1+ b2+ b3)2. Hence associativity is true. Further, (A1, B1, C1) + (0, 0, 0) = (0,0,0) + (A1, B1, C1) = (A1, B1, C1), since (0,0,0) is trivially present in Pij for every i, j = 1 to n. Thus Pij is a commutative monoid for i, j = 1 to n.

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2. PYTHAGOREAN QUADRUPLES IN FUZZY MATRICES 2.1 Definition: Let (A, B, C, D) be in Qij, where Qij = Fij × Fij × Fij × Fij such that their m-norms satisfy the condition d2 = a2 + b2 + c2 where ||A||m = a, ||B||m = b, ||C||m = c and ||D||m = d. Then (A, B, C, D) is a

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Pythagorean quadruple fuzzy matrix.

By defining addition in the same manner, as we do in Pij, Qij also becomes a commutative monoid for i, j = 1 to n. In the same way, it can be extended to n tuples. i.e., If Nij = {(A1, A2, … An) ∈ Fij × Fij × ….. × Fij (n times) | an2 = a12 + a22 + … + an-12}, then Nij

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is also a commutative monoid for all i and j. 3. PROGRESSIONAL MATRICES

3.1 Definition: A sequence of matrices in Fij is said to be an arithmetic progressional matrix if their m-norms form an Arithmetic Progression.

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A. NAGOOR GANI and G. KALYANI

3.2 Definition: A sequence of matrices in Fij is said to be a geometric progressional matrix if their mnorms form a Geometric Progression. In Pij, a norm is introduced in the following way, which we name it as Pythagorean norm or Pnorm.

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3.3 Definition: Let X = (A, B, C) be in Pij. Then the function ||..|| :Pij →R defined as ||X|| = a + b + c, for all X in Pij, where ||A||m = a, ||B||m = b and ||C||m = c satisfies the following conditions. (i) ||X|| ≥ 0 and ||X|| = 0 if and only if X = 0.

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(ii) If k ∈ [0, 1], then ||kX|| = |k| ||X|| = k ||X|| (iii) ||X+Y|| ≤ ||X|| + ||Y||, for all X, Y in Pij.

||X|| = (a + b + c) ≥ 0 is true, since a, b, c belong to [0, 1]. Also ||X|| = 0 ⇒ (a + b + c)

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Proof : (i)

= 0 ⇒ ||A||m = ||B||m = ||C||m = 0 ⇒ X = (A, B, C) = (0, 0, 0). Conversely, if X = 0 then A = B = C = 0. Therefore, ||A||m = ||B||m = ||C||m = 0 ⇒ X = 0. Hence ||X|| = 0 if and only if X = 0. kX = k (A, B, C) = (kA, kB, kC). Therefore, ||kX|| = (ka + kb + kc) = k (a + b + c)

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(ii)

= k ||X|| = |k| ||X|| , since k ∈ [0, 1].

(iii) ||X+Y|| = ||A1, B1, C1) + (A2, B2, C2)|| = ||A1+A2, B1+B2, C1+C2|| = (a1+a2) + (b1+b2) +

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(c1+c2) = (a1 + b1 + c1) + (a2 + b2+ c2). Therefore, ||X + Y|| = ||X|| + ||Y||. Hence ||X + Y|| ≤ ||X|| + ||Y||. In fact, ||X + Y|| = ||X|| + ||Y||.

Note: This function is called Pythagorean norm and is denoted by ||..||p which is shortly known as Pnorm.

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3.4 Definition: d :Pij× Pij → R be defined as d(X,Y) = ||X+Y||P = ||X+Y||. Then d is a metric on Pij. For, (i) d(X,Y) = ||X+Y|| ≥ 0 and if d(X,Y) = 0, then ||X+Y|| = 0. i.e. (a1 + a2 + b1 + b2 + c1 + c2) = 0 ⇒ ai = bi = ci = 0 for i=1, 2, since ai, bi, ci ∈ [0, 1] ⇒ X = 0, Y = 0. Conversely X = 0, Y = 0 ⇒ ai = bi

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= ci = 0 for i=1, 2. Therefore ||X+Y|| = 0 ⇒ d(X,Y) = 0. Hence d(X,Y) = 0 if and only if X = Y. (ii) Clearly d(X,Y) = d(Y, X)

(iii) d(X,Y) + d(Y, Z) = ||X+Y|| + ||Y+Z|| = ||X|| + ||Y|| + ||Y|| + ||Z|| = (a1 + b1 + c1) + (a2 + b2+ c2) + (a2 + b2+ c2) + (a3 + b3+ c3) = (a1 + b1 + c1) + (a3 + b3+ c3) + 2(a2 + b2+ c2) = d(X, Z) + a non-

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negative quantity. Hence d(X,Z) ≤ d(X,Y) + d(Y, Z). Thus d is a metric on Pij. Hence Pij is a metric

[1] [2] [3]

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commutative monoid for i = 1 to n and j = 1 to n.

REFERENCES

De Dickson. History of Theory of Numbers.Vol 2, Chelsea Publishing Company, New York, 1969. S.G. Telang. Number Theory. Tata Mc. Graw hill, 1996. A. Nagoor Gani and G. Kalyani. On fuzzy m-normed matrices. Bulletin of Pure and Applied Sciences, Vol. 22E (1): 1-11, 2003.

[4]

L.A. Zadeh. Fuzzy sets. Inform. Control, 8: 338-353, 1965.

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